SPIE Proceedings [SPIE SPIE Optical Engineering + Applications - San Diego, California, USA (Sunday 12 August 2012)] Quantum Communications and Quantum Imaging X - Bounds on parameter

Bounds on Parameter Estimation Precision for OpticalRanging Systems

Bruce Moision and Baris I. Erkmen

Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Dr., Pasadena, CA 91109

ABSTRACT

Many errors contribute to the accuracy of a ranging system, among them calibration and estimation errors. Inthis paper we address the estimation error contribution of an optical ranging system. We present achievableRMS errors in estimating the phase, frequency, and intensity of a direct-detected intensity-modulated opticalpulse train. For each parameter, the Cramer-Rao-Bound (CRB) is derived and the performance of the MaximumLikelihood estimator is illustrated. Approximations to the CRBs are provided, enabling an intuitive understand-ing of estimator behavior as a function of the signaling parameters. The results are compared to achievable RMSerrors in estimating the same parameters from a sinusoidal waveform in additive white Gaussian noise. Thisestablishes a framework for a performance comparison of radio frequency (RF) and optical science. Comparisonsare made using parameters for state-of-the-art deep-space RF and optical links. Degradations to the achievableerrors due to clock phase noise and detector jitter are illustrated.

Keywords: optical science, estimation theory, quantum-limited sensing, photon-counting detectors, CramerRao bounds

1. INTRODUCTION

The Jet Propulsion Laboratory is currently developing technology to support optical communications with space-craft in deep-space. A deep-space optical communications link can support orders of magnitude larger data ratesthan its radio-frequency (RF) counterpart for the same terminal mass and power.1, 2 The optical link may alsobe used to derive range and doppler estimates, much as its RF counterpart. We would expect that estimates ofthese parameters would similarly see improvements in accuracy relative to the RF case.

A complete ranging protocol must take into account errors accumulated at many layers of the protocol,including calibration errors, relative-timing errors, parameter estimation errors, etc. In this paper we willconcentrate exclusively on the physical layer, specifically, on the parameter estimation accuracy on the downlinkportion of the ranging system. Our analysis in this paper quantifies the performance improvement that may begained by estimating parameters of an intensity modulated optical signal, rather than those of a pure microwave-frequency tone, and provides an intuitive comparison of the physical signal characteristics governing the accuracyin the two cases. Hence our results address the gains in one component of the error budget governing a completeranging system.

Optical links in development at the Jet Propulsion Laboratory utilize intensity-modulation, noncoherentphoton-counting receivers, and carriers in the infrared regime, from 1.0 to 1.5 μm.1 These choices are made tomaximize the power efficiency of the link and take into account constraints of current technology. In this paper,we consider deriving measurements from this optical telemetry link. Throughout we use the shorthand ‘optical’to refer to an intensity-modulated infrared signal received with a noncoherent photon-counting receiver. In thelarger picture, we would like to determine what science one can derive from observing the telemetry link. Wemay separate this into two sub-problems:

c©California Institute of Technology. Government sponsorship acknowledged. Send correspondence [email protected]. The research described in this publication was supported by the DARPA InPho programunder contract PROP.81-17433, and the JPL R&TD Program, and was carried out by the Jet Propulsion Laboratory,California Institute of Technology, under a contract with the National Aeronautics and Space Administration

Invited Paper

Quantum Communications and Quantum Imaging X, edited by Ronald E. Meyers, Yanhua Shih, Keith S. Deacon, Proc. of SPIE Vol. 8518, 85180S · © 2012 SPIE

CCC code: 0277-786/12/$18 · doi: 10.1117/12.930096

Proc. of SPIE Vol. 8518 85180S-1

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1. How accurately can we measure distortions of an intensity-modulated optical signal?

2. What science can be performed given the accuracy of the measurements?

Here we address one component of the first question, determining the accuracy with which one may measurephase, frequency, and intensity of an optical signal. We compare this to the conventional state-of-the-art esti-mation from coherently demodulated microwave, or radio-frequency (RF), carriers. Conventional RF science isderived from a ranging clock, which may be modeled as a sinusoid received in AWGN, see, e.g., [3, Chapter 3],

y(t) = A cos (2πfrt− φ) + n(t) (1)

where A is the signal amplitude, fr is the range clock frequency, φ is the phase, and n(t) is additive white Gaussiannoise. An example is illustrated in Figure 1 for fr = 1 MHz. Estimating the instantaneous phase, frequency, andamplitude of the RF waveform allows one to determine properties of processes that distort these parameters.For example, spacecraft range may be determined from the transmitter delay, via an estimate of the phase, andvelocity determined by the induced doppler shift, via an estimate of the frequency. Analogous properties of thesignal waveform may be extracted from the photocurrent generated from an intensity-modulated optical signalincident on a photon-counting detector. The photocurrent, when normalized by the electron charge, is modeledas a random Poisson point process, governed by a rate function

l(t) = lp

∞∑

k=−∞ρ(t− kTr − φ′) + lb (2)

where lp denotes the peak photo-electron rate in pe/sec, ρ(t) is the intensity pulse shape, satisfying 1 ≤ ρ(t) ≥ 0,Tr is the repetition period, φ′ is the phase, and lb is the mean photo-electron rate from other sources, such asthermal∗ and dark noise †. Figure 2 illustrates an example rate function and a realization of a random pointprocess, representing photo-electron arrivals, induced by it. Analogous to the parameter estimates obtained froman RF carrier, we may estimate lp, Tr, and φ, based on observing photo-electron emissions governed by the ratefunction l(t).

0 1 2 3 4

−1

0

1

t (μs)

y(t)

Figure 1. Coherent Microwave (RF) Received Signal

The paper is organized as follows. In Section 2, we establish a model and general framework for studying theproblem, and derive Cramer-Rao-bounds (CRBs) and maximum-likelihood (ML) estimators. In Section 3 wereview the analogous results for the RF signal and draw comparisons utilizing parameters for current state-of-theart links. In Section 4 we examine degradations to the ideal performance due to detector and clock jitter, andin Section 5 we briefly discuss the results.

2. PARAMETER ESTIMATION FOR A DIRECT-DETECTED OPTICAL SIGNAL

In any implementation of a parameter estimation system, many errors contribute to the overall performance,such as clock drift, clock offset, transmitter jitter and detector jitter. In this section we consider the performance

∗We assume thermal noise is multi-mode with sufficient number of modes to justify a Poisson approximation†Although dark noise does not, strictly speaking, generate photo-electrons, for convenience we utilize units of photo-

electrons for all electrons.



0 0.1 0.2 0.3 0.40

5

10

15

x 106

t (μs)

l(t)

Figure 2. Noncoherent Infrared (Optical) Received Signal

of an ideal system, limited only by the received signal and noise powers and observation times. In Section 4 weintroduce degradation due to detector jitter and clock phase noise. Thus, the performance reported herein is alower bound on the performance of practical systems.

2.1 Channel Model

We model the received signal as follows. A periodic repeating optical pulse train is transmitted in vacuum. Atthe receiver, the light is focused on an ideal photodetector. The ideal detector has negligible thermal noise,and sufficient bandwidth that individual photon arrival times may be observed at its output. The output isa random photocurrent, which, normalized by the electron charge, is accurately modeled as an inhomogeneousPoisson process with rate function given by (2).

The signal parameters are determined by the underlying communications link, which we presume is imple-mented with pulse-position-modulation (PPM), wherein time is divided into slots of duration Ts, with Ts on theorder of the pulse width. It is convenient, in this context, to normalize time to be in units of slots‡, that is,defining

udef= t/Ts , (3)

the rate function of the inhomogenous Poisson process that describes the photocurrent is expressed as

Λ(u)def= Tsl(uTs) = ns

∞∑

k=−∞g(u− kT − φ) + nb , (4)

where nsdef= lp

∫g(t)dt is the mean signal photo-electrons per pulse, nb

def= lbTs is the mean number of noise

photo-electrons per slot, Tdef= Tr/Ts is the repetition period in slots, φ = φ′/Ts is the phase of the periodic

waveform, and

g(u) =f(uTs)∫f(uTs)du

(5)

is the normalized pulse shape. We assume a generalized Gaussian pulse,

g(u) =p

2aΓ(1/p)exp(−|u/a|p) , (6)

where Γ(·) is the Gamma function, a is the 1/e width of the pulse, and p is the decay rate of the pulse tails.Figure 3 illustrates the pulse for a range of values of p. This parameterized pulse shape models a wide range ofpractical pulse shapes, from Gaussian (p = 2), to square (large p, e.g., p = 10 or greater).

We treat the problems of estimating ns, T and φ. We assume throughout that the unknown parameters areconstant over the observation period. We also assume that the noise mean nb is known§. Estimates are based

‡Note that this normalization is simply a unit conversion to units of time in slotwidths at the receiver. This is done tosimplify notation and make it easier to generalize results, but does not imply the receive has knowledge of the slotwidth.

§The noise mean is typically slowly time varying, and may be estimated with an accuracy justifying this assumption.



-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

u

g(u)

p = 2p = 10p = 100

Figure 3. Generalized Gaussian pulse shapes for several decay rates p.

on the observation of photon arrivals over an interval of duration Ti = KT . Suppose that we observe N arrivalsat times {u1, u2, . . . , uN}. The conditional log joint density of the collection of observations is4

log p({ui}, N) =

⎧⎨

⎩− ∫

TiΛ(u)du+

∑N−1i=0 log Λ(ui) ,for N > 0

− ∫TiΛ(u)du ,for N = 0.

(7)

We assume a single parameter is unknown, and all other parameters are known. Let θ ∈ {φ, T, ns} be theunknown parameter. In the following sections we find the maximum-likelihood estimate

θML = argmaxθ

log p({ui}, N ; θ) (8)

and the CRB, which we show in Appendix A may be expressed as

CRBθ =

(∫

Ti

(dΛ(u)/dθ)2

Λ(u)du

)−1

(9)

which, substituting the rate function (4), yields

CRBθ =

(∫

Ti

n2s(∑

k(∂g(u− kT − φ)/∂θ)2

ns

∑k g(u− kT − φ) + nb

du

)−1

(10)

The mean-squared-error of the ML estimator, MSE = E[(θ − θML)2], asymptotically approaches the CRB5 in

Ti, hence the CRB provides a useful approximation to the performance of the estimator in the limit of longobservation times. In the following sections we provide expressions for (8),(10) for each parameter. Throughoutwe assume the pulses are non-overlapping–a good approximation for T � a, which is typically the case.

2.2 Phase

The ML phase estimate is given by

φML = argmaxφ

N−1∑

i=0

log(nsg([ui − φ]T

)+ nb

), (11)



where [u]Tdef= (u+T/2 mod T )−T/2. In this expression, from periodicity of the pulse train, arrivals need only

be known modulo the period. Note that the ML estimator based on observing K periods of the inhomogenousPoisson process with rate Λ(u) is statistically equivalent to that based on observing one period of an inhomogenousPoisson process with rate KΛ(u).

The CRB is given by6

CRBφ =

(K

∫ T/2

−T/2

(nspup−1g(u))2

a2p(nb + nsg(u))du

)−1

(12)

which may be evaluated numerically. In the asymptotic regions we have

CRBφ ≈

⎧⎪⎪⎨

⎪⎪⎩

1

Kns

a2

p, nsg(0) � nb

nb

Kn2s

a222−1/p

p, nsg(0) � nb .

(13)

In our current cases of interest, the signal powers demanded by the communications link put us in the nsg(0) � nb

regime, which we’ll refer to as the high SNR regime. Suppose the time-of-flight of the path is known a-priori towithin one period of the pulse train (the period may be increased by transmitting a pseudo-random sequence ofpulses to make this assumption valid). Then the phase may be mapped to an estimate of the time-of-flight, andto an estimate of the range. If the transmission path is vacuum, the resulting RMS range error (RMS =

√MSE)

is

RMSrange ≈√

1

TiPr

hc

λ

(aTs)2c2

p(m)

where h is Planck’s constant, λ is the carrier wavelength, c is the speed of light in a vacuum, and we’ve mappedthe mean signal photons per pulse to the received power, Pr, as

Pr =nshc

TTsλ(14)

We see that the RMS error is inversely proportional to the square root of the received signal energy, linear inthe pulse width (aTs), and decreases for pulses that are more square (larger p).

2.3 Intensity

The ML estimate of the intensity, ns,ML, satisfies

N−1∑

i=0

g(ui)

ns,MLg(ui) + nb≈ K

If we approximate the pulse as uniform on [−β, β) and zero otherwise, we have the approximation

ns,ML ≈ 1

KN[−β,β] − 2βnb (15)

where N[t1,t2] is the number of arrivals on [t1, t2]. Estimate (15) is the number of arrivals over an approximatepulse duration, minus the mean noise arrivals on the same period. The CRB is given by

CRBns =

(K

∫ T/2

−T/2

g2(u)

nsg(u) + nbdu

)−1

which may be evaluated numerically. In the asymptotic regions, we have

CRBns ≈{ns/K , nsg(0) � nb

(nb/K)Γ(1/p)21/p/p , nsg(0) � nb



To make a common comparison later with the RF case, we convert the estimate of ns to an estimate of thereceived power via (14), with high SNR CRB

CRBPr ≈ Pr

Ti

hc

λ

In practice, we may be more interested in changes to the power, rather than the absolute power, and, hence, thefractional error

RMSPr

Pr≈√(

1

PrTi

)(hc

λ

)

2.4 Period

For the baseband photocurrent process, it is more natural to derive the frequency estimate from an estimate ofthe period T . Let the known phase be zero: φ = 0. Recall that Ti = KT . Let K = KT/T, the number ofpulses that would occur in the interval Ti under the hypothesis T = T . The ML estimate may be written as

TML = argmaxT

J(T )

J(T ) = −ns

K−1∑

k=0

∫ KT

0

g(u− kT )du+

N−1∑

i=0

log Λ(ui; T )

To a good approximation,∫KT

0g(u− kT )du = 1 for all k, hence

J(T ) ≈ −nsK +

N−1∑

i=0

log

⎛

⎝ns

K−1∑

k=0

g(ui − kT ) + nb

⎞

⎠

In the last term, to a good approximation, only one pulse has a non-negligible contribution for each ui, hencewe have the further approximation

J(T ) ≈ −nsK +N−1∑

i=0

log(nsg([ui]T ) + nb) (16)

We find the maximum of (16) numerically, using a grid search around a region of the true value. In doing so,we assume prior knowledge of the parameter domain. This is a valid assumption as the period will be knownwithin a range bounded by uncertainties due to transmitter clock stability and Doppler predicts.

The derivative of the likelihood function is

∂Λ(u)/∂T = ns

K−1∑

k=0

pk

2a2Γ(1 + 1/p)

(u− kT

a

)p−1

exp

(−(u− kM

a

)p)

We assume that pulses are non-overlapping, hence cross terms in the numerator of (10) vanish. Similarly, theintegrand vanishes away from the pulses. Hence we have

CRBT ≈

⎛

⎜⎝n2s

K−1∑

k=1

k2∫ ∞

−∞

(g(u) pa

(ua

)p−1)2

nsg(u) + nbdu

⎞

⎟⎠

−1

=

⎛

⎜⎝(K − 1)(K)(2K − 1)n2

s

6

∫ ∞

−∞

(g(u) pa

(ua

)p−1)2

nsg(u) + nbdu

⎞

⎟⎠

−1

(17)



which may be evaluated numerically. In the asymptotic regions we have

CRBT ≈

⎧⎪⎪⎨

⎪⎪⎩

1

nsK3

3a2Γ(1/p)

p2Γ(2− 1/p)nsg(0) � nb

nb

n2sK

3

24a3Γ2(1/p)

p3Γ(2− 1/p)21/pnsg(0) � nb

(18)

The corresponding CRB for the estimate of the frequency of the pulse train, f = 1/(TTs), in the high SNRregime is

CRBf ≈ 1

PrT 3i

hc

λ

3a2Γ(1/p)

T 2p2Γ(2− 1/p)

For doppler estimation, the relevant error is the fractional error

RMSff

≈√(

1

PrT 3i

)(hc

λ

)(3(aTs)2Γ(1/p)

p2Γ(2− 1/p)

)

3. COMPARISON WITH ESTIMATION FROM A COHERENTLY RECEIVED RFSIGNAL

As discussed earlier, the received RF ranging signal may be modeled as a sinusoid in additive white Gaussiannoise

y(t) = A cos (2πfrt+ φ) + n(t)

where A is the signal amplitude (A =√2Pr where Pr is the received power in the range signal), fr is the

range clock frequency, and φ is the phase. The additive noise n(t) is Gaussian noise with power spectral densitySn(f) = N0/2 Watts/Hz on [−W,W ] and zero elsewhere. The signal is sampled at rate fs = 2W . Put f0 = fr/fs.Estimates are based on a collection of N samples

yn = A cos (2πf0n+ φ) + wn

where wn is an IID, zero-mean, Gaussian sequence with variance σ2 = WN0. ML estimates and the CRBs forestimating (f0, A, φ) are well known, see, e.g., Ref. 5. We simply restate the results here:

CRBφ =2σ2

NA2

φML ≈ arctan

(−∑N−1

n=0 yn sin(2πf0n)∑N−1n=0 yn cos(2πf0n)

)

CRBA =2σ2

N

AML ≈ 4

N

N−1∑

n=0

yn cos(2πf0n+ φ)

CRBf0 ≈ 3σ2

2A2π2N3

f0ML ≈ argmaxf0

N−1∑

n=0

yn cos(2πf0n+ φ)

Table 1 compares the achievable RMS errors (RMS =√MSE ≥ √

CRB), or normalized versions, for idealRF and optical links, using the high signal power asymptotes for the optical case. We see similar behavior foreach parameter. The RMS phase error is inversely proportional to the product of the power and the integrationtime, the frequency error is inversely proportional to the product of the power and the cube of the integrationtime, and the intensity error goes as the power on the integration time. Hence the slopes of RMS error versuseither integration time or SNR will be the same for RF and optical. In order to compare performance requiresa determination of the signal and noise powers, which we treat for a sample pair of links in the next section.



RMS error

parameter optical RF

RMSrange(m) c

√(1

TiPr

)(hc

λ

)(a2T 2

s

p

)c

√(1

TiPr

)(2N0)

(1

(4πfr)2

)

RMSf

f

√(1

T 3i Pr

)(hc

λ

)3(aTs)2Γ(1/p)

p2Γ(2− 1/p)

√(1

T 3i Pr

)(2N0)

(3

(4πfr)2

)

RMSPr

Pr

√(1

PrTi

)(hc

λ

) √(1

PrTi

)(2N0)

Table 1. Comparison of achievable parameter estimation accuracies in the high SNR regime. h is Planck’s constant andc the speed of light in vacuum.

3.1 Representative Link Budgets for a Mars-Earth Downlink

In this section we compare two specific candidate Mars-Earth downlinks: a Ka-band RF link with carrier fre-quency 32 GHz (wavelength 9.3 mm), and an optical link in the near-infrared with carrier frequency 193.5 THz(wavelength 1.55 μm). The link budgets are provided in Table 2. The Ka-band link parameters are chosen to cor-respond to a Ka-bandMars-Reconnaissance-Orbiter link.7, 8 The optical link parameters are chosen to correspondto the Deep-Space-Optical Transceiver (DOT) concept.1 The DOT concept was designed to have comparablemass and power as the Ka-band terminal. Hence the comparison is normalized for comparable burden on thespacecraft terminal. These represent current state-of-the-art candidates for a deep-space telecommunicationslink.

For the optical link, we assume a receive telescope diameter Dr = 11.8 m, corresponding to the LargeBinocular Telescope in southeastern Arizona. We choose Ts = 0.42 ns, a target slotwidth at a range of 0.42 AU.To be conservative, we choose a worst case noise power for this link, Pn = 3.28 pW.1 At long integration times,the noise is negligible, reflected in the large signal power CRBs. From these parameters, we find the receivedpower from the link equation

Pr = Pt

(πDtDr

4Rλ

)2

η

where R is the range and λ the carrier wavelength. From the link budgets we obtain

ns =PrTTsλ

hc= 1.95 (pe/pulse)

nb =PnλTshc

= 0.01 (pe/slot)

For the RF link, we assume a receive antenna diameter Dr = 34 m, corresponding to a Deep Space Networkantenna. We assume a range clock modulation index of 0.8 rad, hence the received power is9

Pr = Pt

(πDtDrfc

4Rc

)2

η2J21 (√2φr)

where J1 is a Bessel function of the first kind or order 1. From the link equation we obtain

A2 = 2Pr = 0.05 (pW)

σ2 = N0W = 0.002 (pW)



Ka-band Linkf0 carrier frequency 32.0 GHzfr range clock 1.0 MHzφr range mod. index 0.8 radφc data mod. index 0.0 radDt transmit diameter 3.0 mDr receiver diameter 34.0 mη system efficiency −10 dBNo noise spectral density −178.45 dB-mW/HzW bandwidth 1.5 MHzPt transmit power 35 W

Near-Infrared Linkλ wavelength 1.55 μmDt transmit diameter 22.0 cmDr receiver diameter 11.8 mη system efficiency −16.74 dBαb noise spatial density 0.03 pW/m2

Ts slot width 0.42 nsp pulse shape 2a pulse width 1/2M = T PPM order 16Pt transmit power 4 W

Table 2. Sample Ka-band and Infrared Link Parameters

3.2 Estimator Performance

Figures 4, 5, and 6 illustrates the ML estimator RMS error and the corresponding CRB for the operating pointsin Table 3. We see a threshold behavior typical of ML estimators: below a certain integration time, the estimatordoes no better than a random guess, above that threshold, it converges rapidly to the CRB.

Since the errors behave the same as a function of the integration time, we see the difference in the RMS errorsmay be factored into three constituent terms: a ratio of received powers, a ratio of noise contributions, and aratio of the bandwidths of the signals. The ratio of the square of the received power for our presumed budgetsis 16 dB. This is a result of the large divergence gain when transmitting at optical wavelengths. The ratio of thenoise contributions relates the shot noise of the optical signal to the thermal noise of the RF signal. As we havefactored the terms, this benefits the RF signal by approximately 8 dB. The final term is the ratio of the signalfeatures, or bandwidth, which appears as (aTs4πfr)

2/p (note that for p = 2, the ratios of the constants is thesame). For our signals, this ratio is approximately 27 dB. Hence we see gains on the order of 35 dB for rangeand fractional frequency estimates, and on the order of 8 dB for the power estimate, which doesn’t benefit fromthe bandwidth gain.

4. DEGRADATIONS DUE TO DETECTOR JITTER AND CLOCK PHASE NOISE

Prior results illustrate the performance of an estimator with ideal subsystems. These represent bounds on theperformance. Practical systems will have losses relative to the ideal. In this section we consider two non-idealitiesfor a photon-counting receiver: detector jitter, and clock phase noise.

4.1 Detector Jitter

Suppose that we observeN photons at the output of the ideal detector over an interval S at times {t1, t2, . . . , tN}.In any practical detector, photo-electrons are produced with a random offset from the time they would have beenproduced by an ideal detector. That is, the observed photo-electron arrival times are {t1 + δ1, t2 + δ2, . . . , ti +δi, . . . , tN + δN}, where the δi are random variables. We denote this additive noise process as detector jitter, and



10−8

10−7

10−6

10−5

10−4

10−310

−4

10−2

100

102

Integration Time (s)

m

opticalRF

Figure 4. Achievable RMS Range Error (m) for Example RF and Optical One-Way Links. Solid line is the ML estimatorperformance, dashed is the CRB asymptote.

10−8

10−7

10−6

10−5

10−410

−2

10−1

100

101


RM

SP/P

opticalRF

Figure 5. Achievable Fractional RMS Power Error for Example RF and Optical One-Way Links. Solid line is the MLestimator performance, dashed is the CRB asymptote, da-dot (indistinguishable from the CRB) is estimator give by (15)

model the δi as independent of one another as well as of the arrival times {ti}, and identically distributed withdistribution

fδ(δ) =1√2πσ2

δ

e−δ2/(2σ2δ )

which is a reasonably good fit to observed jitter densities,10 and we conjecture represents a worst-case jitterof given standard deviation. Table 3 lists the measured jitter standard deviations10 for an InGaAsP photo-multiplier tube (PMT) selected for high detection efficiency, a niobium nitride super-conducting single photondetector (NbN), and an InGaAsP Geiger-mode avalanche-photo-diode (APD), respectively.

Detector σ (ns)InGaAsP PMT 0.9154NbN SSPD 0.02591InGaAsP GM-APD 0.2939

Table 3. Detector Jitter for Candidate Detectors



10−8

10−7

10−6

10−5

10−410

−8

10−6

10−4

10−2

100


RM

Sf/f

opticalRF

Figure 6. Achievable Fractional RMS Frequency Error for Example RF and Optical One-Way Links. Solid line is the MLestimator performance, dashed is the CRB asymptote.

4.2 Clock Phase Noise

In order to convert the photocurrent to photon arrival times, the photocurrent may be thresholded and used totrigger a sampling of a clock. Let

s(t) = sinΨ(t)

= sin(2πfct+ ψ(t))

denote the clock, where Ψ(t) is the phase at time t, fc is the nominal, or target, frequency, and ψ(t) is the clockphase noise. We model the conversion of current to time by assuming the photocurrent triggers a sampling ofthe clock at times ti + δi. The sampling of the clock produces time-tags

Ti =Ψ(ti + δi)

2πfc= ti + δi + x(ti + δi)

The phase noise term, x(t), is commonly modeled as a band-limited Gaussian random process with a power-lawpower spectral density11

Sx(f) =

{1

(2π)2

∑0α=−4 hα+2f

α , |f | ≤ fh

0 , |f | > fh.

Numerical examples in this paper use the published phase noise specifications for the Symmetricom 9961Hybrid Space-Qualified temperature compensated crystal oscillator (TXCO), which we consider as a candidateoscillator for a deep space communications terminal. The single-side-band (SSB) phase noise spectrum of thisclock is specified in Table 4.2. We presume the clock is operated at a nominal frequency fc = 125MHz, and we

frequency (Hz) value (dBc/Hz)10 −70102 −105103 −135104 −150105 −150

Table 4. Symmetricom 9961 SSB phase noise



take the cutoff frequency to be fh = 2fc. The phase jitter contribution is dominated by the white phase noise,with phase jitter

σx =

√

2

∫ fh

10Hz

Sx(f)df

≈ 1 ps

In comparison, the smallest detector jitter contribution, for the NbN detector, is σδ ≈ 26 ps. Hence, for thedetectors and clock we choose for our comparison, degradation due to phase noise is dominated by the detectorjitter.

4.3 Performance: Mitigating Detector Jitter

Figure 7 illustrates fractional RMS frequency errors in the presence of the detector jitter and clock phase noisedescribed above. The CRB and ML estimator for no jitter or clock phase noise (as illustrated in Fig. 6) is alsoshown. All cases include clock phase noise. Hence, the ‘no jitter’ curve has clock phase noise but no detectorjitter, illustrating negligible loss for the specified clock phase noise. The jitter curves use the worst case of theconsidered detectors, the PMT, with jitter standard deviation 0.92 nsec. The curve labeled ’no comp.’ denotesthe performance when the ML estimator for no detector jitter is applied to the jittered data. We see a loss ofapproximately 8 dB in integration time (for the same RMS error) due to the detector jitter.

If the receiver has knowledge of the detector jitter statistics, then this information may be incorporated intothe estimators. The effect of the jitter is to yield an observed arrival rate function

h(u) = fδ � Λ(u)

where � is the convolution operation. Figure 7 illustrates estimator performance with the PMT with thiscompensation, that is, using the proper rate function. We see a gain of approximately 3 dB from modeling andcompensating for the jitter.

10−8

10−7

10−6

10−5

10−6

10−4

10−2


RM

Sf/f

CRBno jitterPMT, comp.PMT, no comp.

Figure 7. Performance in Detector Jitter and Clock Noise, R = 2.0 AU

5. CONCLUSIONS/DISCUSSION

In this paper we derived the CRBs and ML estimators for independent estimation of the phase, period, andintensity of a direct-detected intensity-modulated (optical) signal. We illustrated that the ML estimator ap-proaches the CRB with short data records, allowing one to use the CRB expressions to predict the performanceof the ML estimator in most cases of practical interest. In comparing the optical CRBs to those for the RF



case, we see the forms are analogous in their dependence on the integration time, signal power, noise power, andbandwidth, where the RF analog of shot noise (photon energy) is N0, and the RF analog of bandwidth (thereciprocal of the pulsewidth for optical) is the range clock frequency. Hence the performance difference betweenoptical and RF systems comes down to the realization of these parameters. We provided a sample comparisonfor sets of parameters taken from mass-and-power normalized spacecraft terminals, illustrating several orders ofmagnitude gain improvement in the CRBs for the optical system over its RF counterpart. This addresses onlyone aspect of a full ranging system. A complete comparison will have to look at all error sources, including jointestimator of unknown parameters, and resolution of the period ambiguity, in a full, round-trip, ranging protocol.

APPENDIX A. CRAMER RAO BOUND

Here we derive an expression for the CRB of parameters of the intensity function of a Poisson process. LetΛ(u) be the intensity function of a Poisson process, N (u) the (random) counting process induced by Λ(u), andi(t) = dN (u)/du–in our case, the resulting normalized photo-current (a sum of discrete photo-electrons). Notethat

E(i(u)) = Λ(u)

Let {ui} i = 0, . . . , N − 1, be the collection of arrival times over an observation interval Ti, and θ an unknownparameter characterizing Λ(u). We have

∂

∂θlog p({ui}, N ; θ) = −

∫

Ti

Λ(u)du+

N−1∑

i=0

Λ(ui)

Λ(ui)(19)

where Λ(u) = ∂∂uΛ(u). Let h(u) be some deterministic function of u. Note that

∑

i

h(ui) =

∫i(u)h(u)du

E

[∑

i

h(ui)

]=

∫Λ(u)h(u)du

Applying this to (19) we have

E

[− ∂

∂θlog p({ui}, N ; θ)

]=

∫

Ti

Λ(u)− Λ(u) +Λ2(ui)

Λ(ui)du

=

∫

Ti

Λ2(u)

Λ(u)du

ACKNOWLEDGMENTS

Many thanks to Charles Greenhall, JPL, for discussions on clock noise and for providing software clock noisegenerators.

REFERENCES

[1] Biswas, A., Hemmati, H., Piazzolla, S., Moision, B., Birnbaum, K., and Quirk, K., “Deep-space opticalterminals (DOT) systems engineering,” IPN Progress Report 42–183 (November 2010).

[2] Williams, W. D., Collins, M., Boroson, D. M., Lesh, J., Biswas, A., Orr, R., Schuchman, L., and Sands,O. S., “RF and optical communications: A comparison of high data rate returns from deep space in the 2020timeframe,” Tech. Rep. 2007-214459, NASA/TM (March 2007). available at http://gltrs.grc.nasa.gov.

[3] Yuen, J. H., ed., [Deep Space Telecommunications Engineering ], ch. Receiver Design and PerformanceCharacteristics, Plenum Press, New York (1983).

[4] Snyder, D. L., [Random Point Processes ], Wiley (1975).



[5] Kay, S. M., [Fundamentals of Statistical Signal Processing: Estimation Theory ], Prentice Hall (1993).

[6] Erkmen, B. I. and Moision, B., “Maximum likelihood time-of-arrival estimation of optical pulses via photon-counting photodetectors,” in [Proceedings of IEEE International Symposium on Information Theory ], 1909–1913, IEEE (June 2009).

[7] Williams, W. D., Collins, M., Hodges, R., Orr, R. S., Sands, O. S., Schuchman, L., and Vyas, H., “High-capacity communications from Martian distances,” Tech. Rep. 2007-214415, NASA/TM (December 2007).available at http://gltrs.grc.nasa.gov.

[8] Shambayati, S., Morobito, D., Broder, J. S., Davarian, F., Lee, D., Mendoza, R., Britcliffe, M., and Weinreb,S., “Mars Reconnaissance Orbiter Ka-band (32 GHz) demonstration: Cruise phase operations,”

[9] Kinman, P. W. and Berner, J. B., “Two-way ranging during early mission phase,” Proceedings of the IEEEAerospace Conference (March 2003).

[10] Moision, B. and Farr, W., “Communication limits due to photon detector jitter,” IEEE Photonics TechnologyLetters 20, 715–717 (May 2008).

[11] Rutman, J. and Walls, F. L., “Characterization of frequency stability in precisions frequency sources,”Proceedings of the IEEE 79 (June 1991).



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